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Paper Reference(s)

6666Edexcel GCEPure Mathematics C4

Advanced LevelSpecimen Paper

Time: 1 hour 30 minutes

Materials required for examination Items included with question papersAnswer Book (AB16) NilMathematical Formulae (Lilac)Graph Paper (ASG2)

Candidates may use any calculator EXCEPT those with the facility forsymbolic algebra, differentiation and/or integration. Thus candidates mayNOT use calculators such as the Texas Instruments TI-89, TI-92,Casio CFX-9970G, Hewlett Packard HP 48G.

Instructions to Candidates

In the boxes on the answer book, write the name of the examining body (Edexcel), your centrenumber, candidate number, the unit title (Pure Mathematics C4), the paper reference (6666), yoursurname, other name and signature.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for Candidates

A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.This paper has eight questions.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner. Answers withoutworking may gain no credit.

37

1. Use the binomial theorem to expand 21

)34(�

� x , in ascending powers of x, up to and includingthe term in x3. Give each coefficient as a simplified fraction.

(5)

2. The curve C has equation

13x2 + 13y2 � 10xy = 52.

Find an expression for xy

dd as a function of x and y, simplifying your answer.

(6)

3. Use the substitution x = tan � to show that

41

8d

)1(11

022 ���

�

�

�

�xx

.

(8)

38

4. Figure 1

Figure 1 shows part of the curve with parametric equations

x = tan t, y = sin 2t,22��

��� t .

(a) Find the gradient of the curve at the point P where t = 3� .

(4)

(b) Find an equation of the normal to the curve at P.(3)

(c) Find an equation of the normal to the curve at the point Q where t = 4� .

(2)

y

xO

Turn over39

5. The vector equations of two straight lines are

r = 5i + 3j � 2k + �(i � 2j + 2k) and

r = 2i � 11j + ak + �(�3i � 4j + 5k).

Given that the two lines intersect, find

(a) the coordinates of the point of intersection, (5)

(b) the value of the constant a,(2)

(c) the acute angle between the two lines. (4)

6. Given that

)32()1(111

2 xxx

��

�

� 2)1( xA�

+ )1( x

B�

+ )32( x

C�

,

(a) find the values of A, B and C. (4)

(b) Find the exact value of xxx

xd

)32()1(1112

1

02�

�

�

��

�, giving your answer in the form k + ln a, where

k is an integer and a is a simplified fraction.

(7)

40

7. (a) Given that u = 2x �

81 sin 4x, show that

xu

dd = sin2 2x.

(4)

Figure 2

Figure 2 shows the finite region bounded by the curve y =

x-axis. This region is rotated through 2� radians about the x-a

(b) Using the result in part (a), or otherwise, find the exact v

�

O

y

x

21

x sin 2x, the line x = 4� and the

xis.

alue of the volume generated.(8)

4�x

Turn over41

8. A circular stain grows in such a way that the rate of increase of its radius is inverselyproportional to the square of the radius. Given that the area of the stain at time t seconds isA cm2,

(a) show that tA

dd �

A1 .

(6)

Another stain, which is growing more quickly, has area S cm² at time t seconds. It is given that

tS

dd =

S

t2e2 .

Given that, for this second stain, S = 9 at time t = 0,

(b) solve the differential equation to find the time at which S = 16. Give your answer to2 significant figures.

(7)

END